Classifying space for U(n): Difference between revisions

Revision as of 00:39, 29 May 2016

Revision as of 13:17, 26 July 2018

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The [[total space]] EU(''n'') of the [[universal bundle]] is given by

:<math>EU(n)=\left \{e_1,\ldots,e_n \ : \ (e_i,e_j)=\delta_{ij}, e_i\in \mathcal{H} \right \}.</math>

Here, ''H'' isdenotes a anseparable infinite-dimensional complex Hilbert space, the ''e''_''i'' are vectors in ''H'', and <math>\delta_{ij}</math> is the [[Kronecker delta]]. Note that all such Hilbert spaces are isomorphic, thus, for explicit calculations, one may for example pick the space of square-integrable functions on some real interval. The symbol <math>(\cdot,\cdot)</math> is the [[inner product]] on ''H''. Thus, we have that EU(''n'') is the space of [[orthonormal]] ''n''-frames in ''H''.

The [[group action]] of U(''n'') on this space is the natural one. The [[base space]] is then

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and is the set of [[Grassmannian]] ''n''-dimensional subspaces (or ''n''-planes) in ''H''. That is,

:<math>BU(n) = \{ V \subset \mathcal{H} \ : \ \dim V = n \}</math>

so that ''V'' is an ''n''-dimensional vector space.

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One also has the relation that

:<math>BU(1)= PU(\mathcal{H}),</math>

that is, BU(1) is the infinite-dimensional [[projective unitary group]]. See that article for additional discussion and properties.